Tenney norm: Difference between revisions
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If p/q is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer pq. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the [[Benedetti height]], leading to Tenney [[height]]. In either form it is widely used as a [[measure of inharmonicity]] and/or complexity for intervals. It is also known as ''log product complexity''. | If p/q is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer pq. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the [[Benedetti height]], leading to Tenney [[height]]. In either form it is widely used as a [[measure of inharmonicity]] and/or complexity for intervals. It is also known as ''log product complexity''. | ||
The '''Tenney height''' of a [[monzo]] is given by | The '''Tenney height''' of a [[monzo]] b = {{monzo| ''b''<sub>π (2)</sub> ''b''<sub>π (3)</sub> … ''b''<sub>π (''p'')</sub> }} is given by | ||
< | <math>\lVert W^{-1}b \rVert = \vert b_{\pi (2)} \vert + \log_2 (3) \vert b_{\pi (3)} \vert + \ldots + \log_2 (p) \vert b_{\pi (p)} \vert = \log_2 (2^{|b_{\pi (2)}|} \cdot 3^{|b_{\pi (2)}|} \cdot \ldots \cdot p^{|b_{\pi (p)}|})</math> | ||
where W is the Tenney weighter such that, for the prime basis Q = {{val| 2 3 5 … ''p'' }}, | |||
<math>W = \operatorname {diag} (1/log_2 (Q))</math> | |||
== Examples == | == Examples == | ||